A357196 -id:A357196 - cp's OEIS frontend

A357197 Number of vertices in a hexagon when n internal hexagons are drawn between the 6n points that divide each side into n+1 equal parts.

6, 12, 30, 60, 102, 156, 222, 300, 390, 468, 606, 708, 870, 1020, 1152, 1356, 1542, 1740, 1950, 2112, 2406, 2652, 2910, 3072, 3462, 3756, 4062, 4350, 4710, 4974, 5406, 5772, 6126, 6540, 6918, 7260, 7782, 8220, 8646, 8946, 9606, 10032, 10590, 11052, 11568, 12156, 12702, 13116, 13830, 14388

Offset: 0

Scott R. Shannon, Sep 17 2022

nonn

Unlike similar dissections of the triangle and square, see A357007 and A357060, there is no obvious pattern for n values that yield hexagons with non-simple intersections; these n values begin 9, 11, 14, 19, 23, 27, 29, 32, 34, 35, 38, 39, 41, 43, ... .

Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 9. This is the first term that forms hexagons with non-simple intersections.
Scott R. Shannon, Image for n = 50.
Scott R. Shannon, Image for n = 150.

Cf. A357196 (regions), A357198 (edges), A330846, A357007 (triangle), A357060 (square).

a(n) = A357198(n) - A357196(n) + 1 by Euler's formula.

Conjecture: a(n) = 6*n^2 + 6 for hexagons that only contain simple intersections when cut by n internal hexagons.

A357198 Number of edges in a hexagon when n internal hexagons are drawn between the 6n points that divide each side into n+1 equal parts.

Original entry on oeis.org

6, 18, 54, 114, 198, 306, 438, 594, 774, 942, 1206, 1422, 1734, 2034, 2310, 2706, 3078, 3474, 3894, 4242, 4806, 5298, 5814, 6186, 6918, 7506, 8118, 8706, 9414, 9978, 10806, 11538, 12258, 13074, 13842, 14562, 15558, 16434, 17298, 17970, 19206, 20082, 21174, 22122, 23154, 24306, 25398, 26286

Offset: 0

Scott R. Shannon, Sep 17 2022

nonn

Unlike similar dissections of the triangle and square, see A357008 and A357061, there is no obvious pattern for n values that yield hexagons with non-simple intersections; these n values begin 9, 11, 14, 19, 23, 27, 29, 32, 34, 35, 38, 39, 41, 43, ... .

Cf. A357196 (regions), A357197 (vertices), A330845, A357008 (triangle), A357061 (square).

a(n) = A357196(n) + A357197(n) - 1 by Euler's formula.

Conjecture: a(n) = 12*n^2 + 6 for hexagons that only contain simple intersections when cut by n internal hexagons.

A357216 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of regions in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

Original entry on oeis.org

1, 4, 1, 13, 5, 1, 28, 17, 6, 1, 49, 37, 21, 7, 1, 70, 65, 46, 25, 8, 1, 109, 93, 81, 55, 29, 9, 1, 148, 145, 126, 97, 64, 33, 10, 1, 181, 181, 181, 151, 113, 73, 37, 11, 1, 244, 257, 246, 217, 176, 129, 82, 41, 12, 1, 301, 309, 321, 295, 253, 201, 145, 91, 45, 13, 1

Offset: 3

Scott R. Shannon, Sep 18 2022

Conjecture: the only n-gons that contain non-simple intersections are the 3-gon (triangle), 4-gon (square), and 6-gon (hexagon).

			The table begins:
  1,  4, 13,  28,  49,  70, 109, 148, 181,  244,  301,  334,  433,  508,  565, ...
  1,  5, 17,  37,  65,  93, 145, 181, 257,  309,  401,  457,  577,  653,  785, ...
  1,  6, 21,  46,  81, 126, 181, 246, 321,  406,  501,  606,  721,  846,  981, ...
  1,  7, 25,  55,  97, 151, 217, 295, 385,  475,  601,  715,  865, 1015, 1159, ...
  1,  8, 29,  64, 113, 176, 253, 344, 449,  568,  701,  848, 1009, 1184, 1373, ...
  1,  9, 33,  73, 129, 201, 289, 393, 513,  649,  801,  969, 1153, 1353, 1569, ...
  1, 10, 37,  82, 145, 226, 325, 442, 577,  730,  901, 1090, 1297, 1522, 1765, ...
  1, 11, 41,  91, 161, 251, 361, 491, 641,  811, 1001, 1211, 1441, 1691, 1961, ...
  1, 12, 45, 100, 177, 276, 397, 540, 705,  892, 1101, 1332, 1585, 1860, 2157, ...
  1, 13, 49, 109, 193, 301, 433, 589, 769,  973, 1201, 1453, 1729, 2029, 2353, ...
  1, 14, 53, 118, 209, 326, 469, 638, 833, 1054, 1301, 1574, 1873, 2198, 2549, ...
  1, 15, 57, 127, 225, 351, 505, 687, 897, 1135, 1401, 1695, 2017, 2367, 2745, ...
  1, 16, 61, 136, 241, 376, 541, 736, 961, 1216, 1501, 1816, 2161, 2536, 2941, ...
  ...
See the attached text file for further examples.
See A356984, A357058, A357196 for more images of the n-gons.

Scott R. Shannon, Extended table for n = 3..50, k = 0..75.
Scott R. Shannon, Image of T(5,20) = 2001.
Scott R. Shannon, Image of T(7,10) = 701.

Cf. A357235 (vertices), A357254 (edges), A356984 (triangle), A357058 (square), A357196 (hexagon), A007678, A344857.

T(n,k) = A357254(n,k) - A357235(n,k) + 1 by Euler's formula.
T(n,0) = 1.
T(n,1) = n + 1.
Conjectured formula for all columns for n >= 7: T(n,k) = n*k^2 + 1.
T(3,k) = A356984(k).
T(4,k) = A357058(k).
T(6,k) = A357196(k).
Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = n*k^2 + 1.

A357254 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

Original entry on oeis.org

3, 9, 4, 27, 12, 5, 57, 36, 15, 6, 99, 76, 45, 18, 7, 135, 132, 95, 54, 21, 8, 219, 180, 165, 114, 63, 24, 9, 297, 292, 255, 198, 133, 72, 27, 10, 351, 348, 365, 306, 231, 152, 81, 30, 11, 489, 516, 495, 438, 357, 264, 171, 90, 33, 12, 603, 604, 645, 594, 511, 408, 297, 190, 99, 36, 13

Offset: 3

Scott R. Shannon, Sep 20 2022

Conjecture: the only n-gons that contain non-simple intersections are the 3-gon (triangle), 4-gon (square), and 6-gon (hexagon).

			The table begins:
   3,  9,  27,  57,  99, 135,  219,  297,  351,  489,  603,  645,  867, 1017, ...
   4, 12,  36,  76, 132, 180,  292,  348,  516,  604,  804,  892, 1156, 1284, ...
   5, 15,  45,  95, 165, 255,  365,  495,  645,  815, 1005, 1215, 1445, 1695, ...
   6, 18,  54, 114, 198, 306,  438,  594,  774,  942, 1206, 1422, 1734, 2034, ...
   7, 21,  63, 133, 231, 357,  511,  693,  903, 1141, 1407, 1701, 2023, 2373, ...
   8, 24,  72, 152, 264, 408,  584,  792, 1032, 1304, 1608, 1944, 2312, 2712, ...
   9, 27,  81, 171, 297, 459,  657,  891, 1161, 1467, 1809, 2187, 2601, 3051, ...
  10, 30,  90, 190, 330, 510,  730,  990, 1290, 1630, 2010, 2430, 2890, 3390, ...
  11, 33,  99, 209, 363, 561,  803, 1089, 1419, 1793, 2211, 2673, 3179, 3729, ...
  12, 36, 108, 228, 396, 612,  876, 1188, 1548, 1956, 2412, 2916, 3468, 4068, ...
  13, 39, 117, 247, 429, 663,  949, 1287, 1677, 2119, 2613, 3159, 3757, 4407, ...
  14, 42, 126, 266, 462, 714, 1022, 1386, 1806, 2282, 2814, 3402, 4046, 4746, ...
  15, 45, 135, 285, 495, 765, 1095, 1485, 1935, 2445, 3015, 3645, 4335, 5085, ...
  ...
See the attached text file for further examples.
See A356984, A357058, A357196 for images of the n-gons.

Scott R. Shannon, Extended table for n = 3..50, k = 0..75.

Cf. A357216 (regions), A357235 (vertices), A357008 (triangle), A357061 (square), A357198 (hexagon), A356984, A357058, A357196, A135565, A344899.

T(n,k) = A357216(n,k) + A357235(n,k) - 1 by Euler's formula.
T(n,0) = n.
T(n,1) = 3n.
Conjectured formula for all columns for n >= 7: T(n,k) = 2n*k^2 + n.
T(3,k) = A357008(k).
T(4,k) = A357061(k).
T(6,k) = A357198(k).
Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = 2n*k^2 + n.

cp's OEIS Frontend

A357197 Number of vertices in a hexagon when n internal hexagons are drawn between the 6n points that divide each side into n+1 equal parts.

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A357198 Number of edges in a hexagon when n internal hexagons are drawn between the 6n points that divide each side into n+1 equal parts.

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A357216 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of regions in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

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A357254 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

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